The Mathematics of Heuristic Portfolio Optimization (HPO)

Practitioners allocate capital with forecast-light rules such as equal weight, inverse volatility, risk parity, HRP, and return-adjusted HRP (RA-HRP). This paper develops \emph{Heuristic Portfolio Optimization} (HPO): an information-restricted projection of the Markowitz/tangency solution onto a stable rule class. The implied-return principle, $\mathbf{w}$ is maximum-Sharpe iff $\mathbf{\mu}_e \propto \mathbf{\Sigma}\mathbf{w}$, gives closed-form optimality sets for leading heuristics and exposes the Schur-complement substitutions behind HRP. For RA-HRP, we introduce fixed-tree cluster-Sharpe recursion, unit-free HRP--RA-HRP interpolation, tangency conditions, conditional-risk splits, and pathwise/KL decompositions of weight distortion. First-order Sharpe calculus expresses the marginal value of return information as nodewise alphas against HRP and yields a linear KL trust budget. We formalize generic HPO maps, define the implied-return defect, prove that it equals squared Sharpe inefficiency, characterize tree-HPO coincidence by nodewise mass ratios, and give a bias--variance decomposition for estimated rules. Finally, HPO is embedded into Reinforcement Learning Portfolio Optimization (RLPO): every HPO map induces a deterministic stationary policy; static HPO is the $\gamma=0$ no-friction face of the Bellman problem; RA-HRP supplies a hierarchical policy prior; and dynamic improvement is warranted when continuation value exceeds myopic HPO defect plus frictions. A performance-difference identity prices the myopic value gap, gives an $\varepsilon/(1-\gamma)$ myopia bound, and identifies nodewise alphas as policy-gradient coordinates of the hierarchical actor. Thus HPO is the static optimality layer and RLPO the dynamic control layer. The conditions are GRS-testable, extend to mean--CVaR and expected utility under ellipticity, and become Kelly-growth conditions in diffusion limits.